Question

For the number $2^2 \times 5^2 \times 17^1$, find

(a) Number of divisors (b) Number of even divisors (c) Number of odd divisors

• A. Number of divisors: 18, Number of even divisors: 12, Number of odd divisors: 6
• B. Number of divisors: 18, Number of even divisors: 6, Number of odd divisors: 12
• C. Number of divisors: 12, Number of even divisors: 18, Number of odd divisors: 6
• D. Number of divisors: 6, Number of even divisors: 12, Number of odd divisors: 18

Answer 1

Answer: (A)

(a) Number of divisors: 18 (b) Number of even divisors: 12 (c) Number of odd divisors: 6

Answer 2

Let the number be $N = 2^2 \times 5^2 \times 17^1$.

(a) The number of divisors is found by adding 1 to each exponent and multiplying the results: $(2+1)(2+1)(1+1) = 3 \times 3 \times 2 = 18$.

(b) For an even divisor, the exponent of 2 must be at least 1. So, the exponent of 2 can be 1 or 2 (2 choices). The exponents of 5 and 17 can be anything from 0 up to their respective powers. Number of even divisors = (Number of choices for exponent of 2) $\times$ (Number of choices for exponent of 5) $\times$ (Number of choices for exponent of 17) Number of even divisors = $(2) \times (2+1) \times (1+1) = 2 \times 3 \times 2 = 12$.

(c) For an odd divisor, the exponent of 2 must be 0 (1 choice). The exponents of 5 and 17 can be anything from 0 up to their respective powers. Number of odd divisors = (Number of choices for exponent of 2) $\times$ (Number of choices for exponent of 5) $\times$ (Number of choices for exponent of 17) Number of odd divisors = $(1) \times (2+1) \times (1+1) = 1 \times 3 \times 2 = 6$.

Check: Total divisors = Even divisors + Odd divisors = $12 + 6 = 18$. This matches our calculation for the total number of divisors.